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Overview

In the following notebok we compute the $ \mathrm{TSE} $ and the $ \mathrm{\overline{TSE}} $ from AVISO data. The notebook is structured as follows:

  1. Import data from the file 'AVISO.mat' stored in the folder 'data'.
  2. Define computational parameters (such as the number of cores) and data.
  3. Define spatio-temporal domain.
  4. Interpolate velocity from (discrete) gridded data

  5. Trajectory Stretching Exponent $ \mathrm{TSE} $ and averaged Trajectory Stretching Exponent $ \mathrm{\overline{TSE}} $ :

    • Compute velocity along trajectories $ \lbrace \mathbf{\dot{x}}(t_{i}) \rbrace_{i=0}^N $ over the time interval $ [t_0, t_N] $.

    • Compute $ \mathrm{TSE} $ from velocity along trajectories:

      \begin{equation} \mathrm{TSE}_{t_0}^{t_N} = \dfrac{1}{t_N-t_0} \log{\dfrac{\sqrt{|\mathbf{\dot{x}}(t_N)|^2 +\mathrm{v}_0^2}}{\sqrt{|\mathbf{\dot{x}}(t_0)|^2 +\mathrm{v}_0^2}}} \end{equation}

    • Compute $ \mathrm{\overline{TSE}} $ from velocity along trajectories:

      \begin{equation} \mathrm{\overline{TSE}}_{t_0}^{t_N} = \dfrac{1}{t_N-t_0} \sum_{i=0}^{N-1}|\log{\dfrac{\sqrt{|\mathbf{\dot{x}}(t_{i+1})|^2 +\mathrm{v}_0^2}}{\sqrt{|\mathbf{\dot{x}}(t_i)|^2 +\mathrm{v}_0^2}}}| \end{equation}

      $ \mathrm{v}_0 $ is a characteristic velocity, which can be be estimated by taking the spatio-temporal average of the velocity over the whole dataset.

Import Data

Computational parameters and data

Here we define the computational parameters and the data.

Spatio-temporal domain

Here we define the spatio-temporal domain over which to consider the dynamical system.

Velocity interpolation

In order to evaluate the velocity field at arbitrary locations and times, we must interpolate the discrete velocity data. The interpolation with respect to time is always linear. The interpolation with respect to space can be chosen to be "cubic" or "linear". In order to favour a smooth velocity field, we interpolate the velocity field in space using a cubic interpolant.

Trajectory Stretching Exponent

averaged Trajectory Stretching Exponent ($ \mathrm{\overline{TSE}}$)

The $ \mathrm{\overline{TSE}}_{t_0}^{t_N} $ and the $ \mathrm{TSE}_{t_0}^{t_N} $ are single trajectory indicators for hyperbolic LCS. The ridges of these single trajectory metrics approximate the major ridges of the FTLE field (see FTLE) at a much lower computational cost as there is no need to compute the spatial derivatives involved when evaluating the FTLE-field.

References

[1] Haller, G., Aksamit, N., & Encinas-Bartos, A. P. (2021). Quasi-objective coherent structure diagnostics from single trajectories. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(4), 043131.